Universidade Federal de Viçosa
Programa de pós-graduação em Genética e Melhoramento
Departamento de Biologia Geral
A new look on the genotype-by-environment interaction: enviromics and probabilistic models
Prof. Dr. Kaio Olimpio das Graças Dias
Dr. Saulo Fabrício da Silva Chaves
\[ y_i = \beta_0 + \beta_1 x_i + \epsilon_i \]
\[ \begin{bmatrix} \mathbf X' \mathbf X & \mathbf X' \mathbf Z \\ \mathbf Z' \mathbf X & (\mathbf Z'\mathbf Z + \lambda \mathbf K^{-1}) \end{bmatrix} \begin{bmatrix} \mathbf b \\ \mathbf u \end{bmatrix} = \begin{bmatrix} \mathbf X' \mathbf y \\ \mathbf Z' \mathbf y \end{bmatrix} \]
\[ R = \frac{i \times r \times \sigma_g}{L} \]
\[ \mathbf{y} = \mathbf{1} \mu + \mathbf{X}_1 \mathbf{b} + \mathbf{X}_2 \mathbf{g} + \boldsymbol{\varepsilon} \Rightarrow \mathbf{\bar{y}} = \mathbf{1} \mu + \mathbf{Z} \mathbf{g} + \boldsymbol{\varepsilon} \]
\[ \mathbf{y} = \mathbf{1} \mu + \mathbf{X}_1 \mathbf{b} + \mathbf{X}_2 \mathbf{e} + \mathbf{Z}_1 \mathbf{g} + \mathbf{Z}_1 \mathbf{ge} + \boldsymbol{\varepsilon} \]
\[ \mathbf{y} = \mathbf{1} \mu + \mathbf{g} + \boldsymbol{\varepsilon} \]
with \(\mathbf{g} \sim \mathcal{N}(0, \mathbf{G} \sigma^2_g)\), and \(\mathbf{G} \rightarrow\) VanRaden (2008)
\[ \mathbf{y} = \mathbf{1} \mu + \mathbf{Z}_1 \mathbf{g} + \mathbf{Z}_2 \mathbf{w} + \boldsymbol{\varepsilon} \]
with \(\mathbf{w} \sim \mathcal{N}(0, \mathbf{\Omega} \sigma^2_w)\), and \(\mathbf{\Omega} = \frac{\mathbf{W}^\prime \mathbf{W}}{q}\)
How to address marker x EF interaction?
\[ \{m_1 w_1, m_1 w_2, ..., m_p w_q \} \]
\[ gw_i = g_i \times w_i \]
with
\[ E(gw_i) = E(g_i) \times E(w_i) = 0 \]
\[ Cov(gw_i, gw_{i^\prime}) = G_{ii^\prime} \times \Omega_{ii^\prime} \]
\[ \mathbf{y} = \mathbf{1} \mu + \mathbf{Z}_1 \mathbf{g} + \mathbf{Z}_2 \mathbf{w} + \mathbf{Z}_3 \mathbf{gw} + \boldsymbol{\varepsilon} \]
with \(\mathbf{gw} \sim \mathcal{N}(\mathbf{0}, \mathbf{G} \# \mathbf{\Omega} \sigma^2_{gw})\)
\[ \mathbf{Z}_1^{(n \times v)} \times \mathbf{G}^{(v \times v)} \times \mathbf{Z}_1^{\prime \, {(v \times n)}} = \mathbf{K}^{(n \times n)} \]
\[ \mathbf{Z}_2^{(n \times v)} \times \mathbf{\Omega}^{(v \times v)} \times \mathbf{Z}_2^{\prime \, {(v \times n)}} = \mathbf{L}^{(n \times n)} \]
\[ \mathbf{L}^{(n \times n)} \# \mathbf{K}^{(n \times n)} \]
\[ \mathbf{y} = \mathbf{1} \mu + \mathbf{Z}_1 \mathbf{g} + \mathbf{Z}_2 \mathbf{w} + \mathbf{Z}_3 \mathbf{gw} + \mathbf{Z}_4 \mathbf{e} + \mathbf{Z}_5 \mathbf{ge} + \boldsymbol{\varepsilon} \]
with \(\mathbf{ge} \sim \mathcal{N}(\mathbf{0}, \mathbf{K} \# \mathbf{Z}_4 \mathbf{Z}_4^\prime)\)
\[ r_w = \frac{\sum_{j=1}^J{\frac{\rho_{\hat{y}\overline{y}{_j}}}{V(\rho_{\hat{y}\overline{y}{_j}})}}}{\sum_{j=1}^J{\frac{1}{V(\rho_{\hat{y}\overline{y}{_j}})}}} \]